arXiv research

Simply-connected surfaces of general type for n≥5.

The problem of finding all minimal surfaces presented in parametric form as polynomials of certain degree is discussed by many authors. It is known that the classical Enneper surface is (up to position in space and homothety) the only polynomial minimal surface of degree 3 in isothermal parameters. In higher degrees th…

For surface-knots, branched covers with degree 3 have simplifying numbers <3.

The Schwarzian derivative helps classify minimal surfaces by their degree.

A new simple proof for surface map degree inequality.

Study shows surfaces with many twistor lines can't be odd-degree.

Maps between surfaces have degree constraints based on their Euler characteristics.

The motivation for this paper is to justify a remark of Thurston that the algebraic degree of stretch factors of pseudo-Anosov maps on a surface $S$ can be as high as the dimension of the Teichmüller space of $S$. In addition to proving this, we completely determine the set of possible algebraic degrees of pseudo-Anoso…

Geometric triangulations of surfaces with uniform vertex degree are shown.

The paper studies knitted surfaces and surface-links, showing their isotopy and closure properties.

Invariant minimal surfaces in the real special linear group of degree 2 with canonical Riemannian and Lorentzian metrics are studied. Constant mean curvature surfaces with vertically harmonic Gauß map are classified.

Minimal maps from surfaces to torus found for various genus values.

Gauss map of complete minimal surfaces avoids certain hypersurfaces.

Characterizes Wahl singularities in del Pezzo surface degenerations.

Finite-type surfaces have a topological Hopf property.

We consider the family of the Bour's minimal surfaces in Euclidean 3-space, and compute their classes, degrees and integral free representations.

In-degree quiver polynomials for surface-links computed.

Study on surfaces containing twistor lines, proving their density and bounds.

For a branched cover between two closed orientable surfaces, the Riemann-Hurwitz formula relates the Euler characteristics of the surfaces, the total degree of the cover, and the total length of the partitions of the degree given by the local degrees at the preimages of the branching points. A very old problem asks whe…

We apply Nadel's method of multiplier ideal sheaves to show that every complex del Pezzo surface of degree at most six whose automorphism group acts without fixed points has a Kähler-Einstein metric. In particular, all del Pezzo surfaces of degree $4,5$, or $6$ and certain special del Pezzo surfaces of lower degree are…

In this paper we study the problem of existence of orbifold Kaehler-Einstein metrics on del Pezzo surfaces of degree 1 with Du Val singular points. Moreover we compute global log canonical thresholds of del Pezzo surfaces of degree 1 with Du Val singularities and of del Pezzo surfaces of Picard rank 1 with Du Val singu…

Study integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.

To a branched cover between closed, connected and orientable surfaces one associates a "branch datum", which consists of the two surfaces, the total degree d, and the partitions of d given by the collections of local degrees over the branching points. This datum must satisfy the Riemann-Hurwitz formula. A "candidate su…

New inequality for odd-degree flexible curves using surface doubling.

Study on roots of Dehn twists on nonorientable surfaces.

A complete description of the global monodromy of a Lefschetz fibration arising from the Fermat surface of degree 4 is given. As a by-product we get a positive relation among right hand Dehn twists in the mapping class group of a closed orientable surface of genus 3.

For a given branched covering between closed connected surfaces, there are several easy relations one can establish between the Euler characteristics of the surfaces, their orientability, the total degree, and the local degrees at the branching points, including the classical Riemann-Hurwitz formula. These necessary re…

Study various complexity measures of curves on surfaces.

Invariants for surfaces up to rigid transformations, with a comeagre subset retrieval algorithm.

New partial solution to Hurwitz problem for surface branched covers.

This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded. In a previous paper by the authors, the problem of calculating whether a given *-graph in which all vertices have degree 4 or 6 admi…

A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In…

Real del Pezzo surfaces split real lines into elliptic and hyperbolic types.

Solves Demailly's system for direct sums of ample line bundles on Riemann surfaces.

Let $F',F$ be any two closed orientable surfaces of genus $g'>g\ge 1$, and $f:F\to F$ be any pseudo-Anosov map. Then we can "extend" $f$ to be a pseudo-Anosov map $f':F'\to F'$ so that there is a fiber preserving degree one map $M(F',f')\to M(F,f)$ between the hyperbolic surface bundles. Moreover the extension $f'$ can…

In this paper we study infinitesimal and finite flexibility for generic semidiscrete surfaces. We prove that generic 2-ribbon semidiscrete surfaces have one degree of infinitesimal and finite flexibility. In particular we write down a system of differential equations describing isometric deformations in the case of exi…

Weierstrass representation is a classical parameterization of minimal surfaces. However, two functions should be specified to construct the parametric form in Weierestrass representation. In this paper, we propose an explicit parametric form for a class of parametric polynomial minimal surfaces of arbitrary degree. It …

Algorithm creates polynomials for knotted surfaces, with bounds on degree.

Paper disproves a Farb question about pseudo-Anosov homeomorphisms.

Given two closed orientable surfaces, the Hurwitz existence problem asks whether there exists a branched cover between them having prescribed global degree and local degrees over the branching points. The Riemann-Hurwitz formula gives a necessary condition, which was shown to be also sufficient when the base surface ha…

Segre quartic surfaces linked to minitwistor spaces with Einstein-Weyl structures.

We show for an alternating knot the minimal boundary slope of an essential spanning surface is given by the signature plus twice the minimum degree of the Jones polynomial and the maximal boundary slope of an essential spanning surface is given by the signature plus twice the maximum degree of the Jones polynomial. For…

As it is well-known, all Vassiliev invariants of degree one of a knot $K\subset R^3$ are trivial. There are nontrivial Vassiliev invariants of degree one, when the ambient space is not $R^3$. Recently, T. Fiedler introduced such invariants of a knot in an $R^1$-fibration over a surface $F$. They take values in the free…

Study links K-stability of certain surfaces to binary forms, proving stability and non-stability conditions.

Researchers compute specific Hurwitz numbers for branched covers.

A $triangulation$ is an embedding of a graph on surfaces where every face has length three. In this article, we show the existence of contractible Hamiltonian cycle in triangulated maps of which minimum degree is four.

We explicitly construct pseudo-Anosov maps on the closed surface of genus $g$ with orientable foliations whose stretch factor $λ$ is a Salem number with algebraic degree $2g$. Using this result, we show that there is a pseudo-Anosov map whose stretch factor has algebraic degree $d$, for each positive even integer $d$ s…

New proof shows surfaces can have identical length spectra but not simple ones.